Kernel Documentation

Kernel Documentation

The research projects, developed by the LAMFO members, tests the application of several different types of Kernel methodology to classify the stocks on Brazilian stock market. This section has the objective to describe the characteristics of each Kernel used and their parameter restriction, being useful for future projects and source of knowledge for those interested in the application and use of it.

Types of Kernels used:

1. Cauchy Kernel
2. Chi-Square Kernel
3. Exponential Kernel
4. Gaussian Kernel
5. Generalized T-Student Kernel
6. Histogram Intersection Kernel
7. Hyperbolic Tangent (Sigmoid) Kernel
8. Inverse Multiquadratic Kernel
9. Laplacian Kernel
10. Linear Kernel
11. Log Kernel
12. Multiquadratic Kernel
13. Power Kernel
14. Rational Quadratic Kernel
15. Wavelet Kernel

Cauchy Kernel

The Cauchy Kernel has it principal parameter restriction on the , used in other Kernels as well to set the density of the normal distribution. Sanyang and Kaban (2014) recommend the use of this Kernel to analyse heavy tail data, identifying observation far from the mean of the distribution.

Parameters Restriction:

• 

• 

Chi-Square Kernel

Different from other Kernel methodology, Chi-Square Kernel doesn't have a parameter restriction or optimization. The only sensitive mater is related with the sample size [(Source)] (https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week05/metcj702_W05S02T05_limitations.html) :

1. This methodology doesn't give information about the relationship or significance of the observations in the sample;
2. Sample size sensitive, the Chi-Square is proportional with the size but has no relation with other variables;
3. Frequency sensitive, specially with small frequency.

Restrictions:

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Exponential Kernel

The Exponential Kernel follow the base of the Gaussian Kernel, normalizing the distribution. The difference between them is the , that represents the density.

Parameters Restriction:

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Gaussian Kernel

Is the most common Kernel used because of the normalization of the data, used as well for the Exponential Kernel. The main issue is the density(), that makes the normal distribution softer than the Exponential Kernel, but it doesn't mean that the classification of the data will be more precise since the bigger the , smaller the amplitude.

Parameters Restriction:

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Generalized T-Student Kernel

Just like the Linear Kernel, the main parameter used on Generalized T-Student Kernel is the acceptance of error or "cost" of the model (d). This "cost" represent the tolerance for miss classification, the bigger the cost more miss classification is accepted.

Parameters Restriction:

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• : known as Hardmargin, there is no acceptance of miss classification;

• : known as Softmargin, there is space for miss classification ().

Histogram Intersection Kernel

The Histogram Intersection Kernel usually is used to image recognittion and doesn't have parameter restrictions or sensibility to sample size, however, this methodology only works for positive data Boughorbel, S.J-P and Nozha, B (2005).

Restriction:

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Hyperbolic Tangent (Sigmoid) Kernel

Used for non-linear classification, this characteristic is granted by the tanhance parameter and the cost the model Lin, H.T and Lin, C.J (2003)he Inverse Multiquadratic Kernel has a unique characteristic).

Parameters Restriction:

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• 

• 

"N" represents the dimention of the data

Inverse Multiquadratic Kernel

The Inverse Multiquadratic has a unique characteristic to generate a full rank matrix based on the data set. The number of columns and rows don't have a superior limit, this feature has an impact over the number of dimensions created to classification, being able to expand until the infinite.

Parameters Restriction:

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Laplacian Kernel

The Laplacian Kernle ues the same base of data normalization as Gaussian and Exponential Kernel, however it is less sensitive to the change on sigma () parameter.

Parameters Restriction:

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Linear Kernel

Is the simplest Kernel since it uses just the "cost" as sensitive parameter.

Parameters Restriction:

• 

• : known as Hardmargin, there is no acceptance of miss classification;

• : known as Softmargin, there is space for miss classification ().

Log Kernel

The Log Kernel has the same principle used on Power Kernel (Boughorbel, S et al., 2005). The idea is to use the "cost" of the model as a dimension parameter, since the data set isn't modified by any other function or parameter.

Parameters Restriction:

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Multiquadratic Kernel

The Multiquadratic Kernel is usually used to analyse data sets with negative observations, using the same base of normalization as Gaussian and Rational Quadratic Kernel. The only parameter used is the "cost" of the model (tolerance of miss classification).

Parameters Restriction:

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Power Kernel

As described in Log Kernel, the Power Kernel only has the "cost" parameter as sensitive key (Boughorbel, S et al., 2005).

Parameters Restriction:

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Rational Quadratic Kernel

The base on normalize the data is used in this Kernel, the difference between the Rational Quadratic Kernel and the other ones, like Gaussian and Multiquadric, is the computational cost os the algorithm. The reason for the performance is related with the intensity of this methodology.

Parameters Restriction:

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Wavelet Kernel

This Kernel is based on Wavelet Theory and has several parameters. The main equation used in this Kernel is:

Parameters Restriction:

• 

• This parameter is known as the translation factor

• This parameter is known as the dilatation factor

The is the base of the Wavelet equations, calculated by:

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For more information, check out the lab website (LAMFO). Other Sources